An extension of mean-variance hedging to the discontinuous case

Arai, Takuji

doi:10.1007/s00780-004-0136-5

Cited by 47 publications

(93 citation statements)

References 9 publications

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“…By contrast, quadratic hedging with discontinuous processes under an arbitrary measure may lead to negative "prices" or not have a solution in general [2].…”

Section: Introductionmentioning

confidence: 98%

“…A natural question is therefore to examine what becomes of the above assertions in presence of discontinuities in asset prices. It is known that, except in very special cases [25], martingales with respect to the filtration of a discontinuous process X cannot be represented in the form (2), leading to market incompleteness. Far from being a shortcoming of models with jumps, this property corresponds to a genuine feature of real markets: the impossibility of "replicating" an option by trading in the underlying asset.…”

Section: Introductionmentioning

confidence: 99%

“…The expectation in (3) can be understood either as being computed under an "objective" measure meant as a statistical model of price fluctuations [2,17,19,27] or as being computed under a martingale ("risk-adjusted") measure [6,7,16,18,24]. Whereas the first choice may seem more natural, there are practical and theoretical motivations for using a risk-adjusted (martingale) measure fitted to market prices of options for computing the hedging performance.…”

Section: Introductionmentioning

confidence: 99%

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Hedging with Options in Models with Jumps

Cont

Tankov

Voltchkova

2007

Stochastic Analysis and Applications

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Abel Symposium 2005 on Stochastic analysis and applications in honor of Kiyosi Ito's 90th birthday. AbstractWe consider the problem of hedging a contingent claim, in a market where prices of traded assets can undergo jumps, by trading in the underlying asset and a set of traded options. We give a general expression for the hedging strategy which minimizes the variance of the hedging error, in terms of integral representations of the options involved. This formula is then applied to compute hedge ratios for common options in various models with jumps, leading to easily computable expressions. The performance of these hedging strategies is assessed through numerical experiments.

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“…By contrast, quadratic hedging with discontinuous processes under an arbitrary measure may lead to negative "prices" or not have a solution in general [2].…”

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hedging with Options in Models with Jumps

Cont

Tankov

Voltchkova

2007

Stochastic Analysis and Applications

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“…Exceptions include [22], which considers the problem of local risk minimization for a model with jumps under the assumption that the stochastic intensity is independent of the processes driving the stock price processes, and the recent paper by Arai [1], which generalizes the methods in [30] to the discontinuous case. Some key differences between [1] and this paper include the generality of the price processes (discontinuous semimartingales v's processes driven by Brownian motion and a doubly stochastic Poisson process) and the methods that are used to solve the problem (duality v's stochastic control). A key issue in both [1] and this paper concerns the so-called variance optimal (signed) martingale measure.…”

Section: Introductionmentioning

confidence: 99%

“…Some key differences between [1] and this paper include the generality of the price processes (discontinuous semimartingales v's processes driven by Brownian motion and a doubly stochastic Poisson process) and the methods that are used to solve the problem (duality v's stochastic control). A key issue in both [1] and this paper concerns the so-called variance optimal (signed) martingale measure. In the continuous semimartingale case, the variance optimal signed martingale measure is actually a probability measure, but this is not necessarily the case when there are jumps.…”

Section: Introductionmentioning

confidence: 99%

Mean-Variance Hedging When There Are Jumps

Lim¹

2005

SIAM J. Control Optim.

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Abstract. In this paper, we consider the problem of mean-variance hedging in an incomplete market where the underlying assets are jump diffusion processes which are driven by Brownian motion and doubly stochastic Poisson processes. This problem is formulated as a stochastic control problem, and closed form expressions for the optimal hedging policy are obtained using methods from stochastic control and the theory of backward stochastic differential equations. The results we have obtained show how backward stochastic differential equations can be used to obtain solutions to optimal investment and hedging problems when discontinuities in the underlying price processes are modeled by the arrivals of Poisson processes with stochastic intensities. Applications to the problem of hedging default risk are also discussed. 1. Introduction. Much of the literature on asset price modeling is motivated by the observation that simple models, like Black-Scholes, fail to account for important features of price processes that are observed in data. For example, the log-returns process of real-world asset prices are not normally distributed but exhibit higher peaks and heavier tails, implying a greater probability of extreme price movements than predicted by Black-Scholes. In addition, the price processes of real-world assets are typically not continuous, but may jump (in a nonpredictable way) in response to news or other surprise events. For a number of years, researchers have focused on developing a richer class of asset price models that include jumps as well as stochastic parameters; see, for example, [3,12,20]. While the adoption of these models in asset pricing (where simulation can be used) is fairly widespread, their use in dynamic optimization problems like hedging and optimal investment, when the market is incomplete, has been quite limited. This paper is concerned with the problem of dynamic mean-variance hedging in an incomplete market when there are random parameters and discontinuities in the price processes. We assume that uncertainty is modeled by Brownian motion and a doubly stochastic Poisson process with intensity that is predictable with respect to the Brownian filtration. We derive expressions for the optimal hedging strategy using methods from stochastic control and the theory of backward stochastic differential equations (BSDEs).While the theory of BSDEs has played an important role in the analysis and solution of mean-variance hedging problems with random parameters (see [25,27]), it is typically assumed that price processes are continuous and driven by Brownian motion. (We note, however, that generalizations to the continuous semimartingale setting have recently appeared; see Bobrovnytska and Schweizer [6].) One contribution of this paper is to show how BSDEs can be used when there are jumps. In this regard,

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Mean-Variance Hedging

Eberlein

Kallsen

2019

Springer Finance

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An extension of mean-variance hedging to the discontinuous case

Cited by 47 publications

References 9 publications

Hedging with Options in Models with Jumps

Hedging with Options in Models with Jumps

Mean-Variance Hedging When There Are Jumps

Mean-Variance Hedging

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